#### Chapters

## Chapter 7: Probability

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 7 Probability Exercise 7.1 [Pages 99 - 100]

State the sample space and n(S) for the following random experiment.

A coin is tossed twice. If a second throw results in a tail, a die is thrown.

State the sample space and n(S) for the following random experiment.

A coin is tossed twice. If a second throw results in head, a die thrown, otherwise a coin is tossed.

In a bag, there are three balls; one black, one red, and one green. Two balls are drawn one after another with replacement. State sample space and n(S).

A coin and a die are tossed. State sample space of following event.

A: Getting a head and an even number.

A coin and a die are tossed. State sample space of following event.

B: Getting a prime number.

A coin and a die are tossed. State sample space of following event.

C: Getting a tail and perfect square.

Find total number of distinct possible outcomes n(S) of the following random experiment.

From a box containing 25 lottery tickets any 3 tickets are drawn at random.

Find total number of distinct possible outcomes n(S) of the following random experiment.

From a group of 4 boys and 3 girls, any two students are selected at random.

Find total number of distinct possible outcomes n(S) of the following random experiment.

5 balls are randomly placed into 5 cells, such that each cell will be occupied.

Find total number of distinct possible outcomes n(S) of the following random experiment.

6 students are arranged in a row for a photograph.

Two dice are thrown. Write favourable Outcomes for the following event.

P: Sum of the numbers on two dice is divisible by 3 or 4.

Two dice are thrown. Write favourable outcomes for the following event.

Q: Sum of the numbers on two dice is 7.

Two dice are thrown. Write favourable outcomes for the following event.

R: Sum of the numbers on two dice is a prime number.

Also, check whether Events P and Q are mutually exclusive and exhaustive.

Two dice are thrown. Write favourable outcomes for the following event.

R: Sum of the numbers on two dice is a prime number.

Also, check whether Events Q and R are mutually exclusive and exhaustive.

A card is drawn at random from an ordinary pack of 52 playing cards. State the number of elements in the sample space if consideration of suits is not taken into account.

A card is drawn at random from an ordinary pack of 52 playing cards. State the number of elements in the sample space if consideration of suits is taken into account.

Box-I contains 8 red (R_{11}, R_{12}, R_{13}) and 2 blue (B_{11}, B_{12}) marbles while Box-II contains 2 red(R_{21}, R_{22}) and 4 blue (B_{21}, B_{22}, B_{23}, B_{24}) marbles. A fair coin is tossed. If the coin turns up heads, a marble is chosen from Box-I; if it turns up tails, a marble is chosen from Box-II. Describe the sample space.

Consider an experiment of drawing two cards at random from a bag containing 4 cards marked 5, 6, 7, and 8. Find the sample Space if cards are drawn with replacement.

Consider an experiment of drawing two cards at random from a bag containing 4 cards marked 5, 6, 7, and 8. Find the sample Space if cards are drawn without replacement.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 7 Probability Exercise 7.2 [Pages 102 - 103]

A fair die is thrown two times. Find the chance that product of the numbers on the upper face is 12.

A fair die is thrown two times. Find the chance that sum of the numbers on the upper face is 10.

A fair die is thrown two times. Find the chance that Sum of the numbers on the upper face is at least 10.

A fair die is thrown two times. Find the chance that sum of the numbers on the upper face is at least 4.

A fair die is thrown two times. Find the chance that the first throw gives an odd number and second throw gives multiple of 3.

A fair die is thrown two times. Find the chance that both the times die shows same number (doublet).

Two cards are drawn from a pack of 52 cards. Find the probability that both are black.

Two cards are drawn from a pack of 52 cards. Find the probability that both are diamonds.

Two cards are drawn from a pack of 52 cards. Find the probability that both are ace cards.

Two cards are drawn from a pack of 52 cards. Find the probability that both are face cards.

Two cards are drawn from a pack of 52 cards. Find the probability that one is spade and other is non-spade.

Two cards are drawn from a pack of 52 cards. Find the probability that both are from same suit.

Two cards are drawn from a pack of 52 cards. Find the probability that both are from same denomination.

Four cards are drawn from a pack of 52 cards. Find the probability that 3 are king and 1 is jack.

Four cards are drawn from a pack of 52 cards. Find the probability that all the cards are from different suit.

Four cards are drawn from a pack of 52 cards. Find the probability that at least one heart.

Four cards are drawn from a pack of 52 cards. Find the probability that all cards are club and one of them a jack.

A bag contains 15 balls of three different colours: Green, Black and Yellow. A ball is drawn at random from the bag. The probability of green ball is `1/3`. The probability of yellow ball is `1/5`. What is the probability of blackball?

A bag contains 15 balls of three different colours: Green, Black and Yellow. A ball is drawn at random from the bag. The probability of green ball is `1/3`. The probability of yellow ball is `1/5`. How many balls are green, black, and yellow?

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. What is the probability that, number on ticket is divisible by 6?

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. What is the probability that, number on ticket is a perfect square?

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. What is the probability that, number on ticket is prime?

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. What is the probability that, number on ticket is divisible by 3 and 5?

From a group of 8 boys and 5 girls, a committee of five is to be formed. Find the probability that the committee contains 3 boys and 2 girls.

From a group of 8 boys and 5 girls, a committee of five is to be formed. Find the probability that the committee contains at least 3 boys.

A room has three sockets for lamps. From a collection of 10 light bulbs of which 6 are defective, a person selects 3 bulbs at random and puts them in socket. What is the probability that the room is lit?

The letters of the word LOGARITHM are arranged at random. Find the probability that vowels are always together.

The letters of the word LOGARITHM are arranged at random. Find the probability that vowels are never together.

The letters of the word LOGARITHM are arranged at random. Find the probability that Exactly 4 letters between G and H.

The letters of the word LOGARITHM are arranged at random. Find the probability that Begins with O and ends with T.

The letters of the word LOGARITHM are arranged at random. Find the probability that start with vowel and end with ends with consonant.

The letters of the word SAVITA are arranged at random. Find the probability that vowels are always together.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 7 Probability Exercise 7.3 [Page 104]

Two dice are thrown together. What is the probability that sum of the numbers on two dice is 5 or number on the second die is greater than or equal to the number on the first die?

A card is drawn from a pack of 52 cards. What is the probability that, card is either red or black?

A card is drawn from a pack of 52 cards. What is the probability that card is either red or face card?

Two cards are drawn from a pack of 52 cards. What is the probability that, both the cards are of same colour?

Two cards are drawn from a pack of 52 cards. What is the probability that, both the cards are either black or queens?

A bag contains 50 tickets, numbered from 1 to 50. One ticket is drawn at random. What is the probability that, number on the ticket is a perfect square or divisible by 4?

A bag contains 50 tickets, numbered from 1 to 50. One ticket is drawn at random. What is the probability that, number on the ticket is a prime number or greater than 30?

Hundred students appeared for two examinations. 60 passed the first, 50 passed the second and 30 passed in both. Find the probability that student selected at random passed in at least one examination.

Hundred students appeared for two examinations. 60 passed the first, 50 passed the second, and 30 passed in both. Find the probability that student selected at random passed in exactly one examination.

Hundred students appeared for two examinations. 60 passed the first, 50 passed the second, and 30 passed in both. Find the probability that student selected at random failed in both the examinations.

If P(A) = `1/4`, `"P"("B") = 2/5` and `"P"("A" ∪ "B") = 1/2` Find the value of the following probability: P(A ∩ B)

If P(A) = `1/4`, P(B) = `2/5` and P(A ∪ B) = `1/2` Find the value of the following probability: P(A ∩ B')

If P(A) = `1/4`, P(B) = `2/5` and P(A ∪ B) = `1/2` Find the value of the following probability: P(A' ∩ B)

If P(A) = `1/4`, P(B) = `2/5` and P(A ∪ B) = `1/2` Find the value of the following probability: P(A' ∪ B')

If P(A) = `1/4`, P(B) = `2/5` and P(A ∪ B) = `1/2` Find the value of the following probability: P(A' ∩ B')

A computer software company is bidding for computer programs A and B. The probability that the company will get software A is `3/5`, the probability that the company will get software B is `1/3`, and the probability that the company will get both A and B is `1/8`. What is the probability that the company will get at least one software?

A card is drawn from a well shuffled pack of 52 cards. Find the probability of it being a heart or a queen.

In a group of students, there are 3 boys and 4 girls. Four students are to be selected at random from the group. Find the probability that either 3 boys and 1 girl or 3 girls and 1 boy are selected.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 7 Probability Exercise 7.4 [Pages 107 - 108]

Two dice are thrown simultaneously, If at least one of the dice show a number 5, what is the probability that sum of the numbers on two dice is 9?

A pair of dice is thrown. If sum of the numbers is an even number, what is the probability that it is a perfect square?

A box contains 11 tickets numbered from 1 to 11. Two tickets are drawn at random with replacement. If the sum is even, find the probability that both the numbers are odd.

A card is drawn from a well-shuffled pack of 52 cards. Consider two events A and B as

A: a club 6 card is drawn.

B: an ace card 18 drawn.

Determine whether the events A and B are independent or not.

A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, problem is solved?

A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, problem is not solved

A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, exactly two students solve the problem?

The probability that a 50-year old man will be alive till age 60 is 0.83 and the probability that a 45-year old woman will be alive till age 55 is 0.97. What is the probability that a man whose age is 50 and his wife whose age is 45 will both be alive after 10 years?

In an examination, 30% of students have failed in subject I, 20% of the students have failed in subject II and 10% have failed in both subject I and subject II. A student is selected at random, what is the probability that the student has failed in subject I, if it is known that he is failed in subject II?

In an examination, 30% of students have failed in subject I, 20% of the students have failed in subject II and 10% have failed in both subject I and subject II. A student is selected at random, what is the probability that the student has failed in at least one subject?

In an examination, 30% of students have failed in subject I, 20% of the students have failed in subject II and 10% have failed in both subject I and subject II. A student is selected at random, what is the probability that the student has failed in exactly one subject?

One-shot is fired from each of the three guns. Let A, B, and C denote the events that the target is hit by the first, second and third guns respectively. assuming that A, B, and C are independent events and that P(A) = 0.5, P(B) = 0.6, and P(C) = 0.8, then find the probability that at least one hit is registered.

A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that, first is white and second is black?

A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that, one is white and other is black?

An urn contains 4 black, 5 white, and 6 red balls. Two balls are drawn one after the other without replacement, What is the probability that at least one ball is black?

Two balls are drawn from an urn containing 5 green, 3 blue, and 7 yellow balls one by one without replacement. What is the probability that at least one ball is blue?

A bag contains 4 blue and 5 green balls. Another bag contains 3 blue and 7 green balls. If one ball is drawn from each bag, what is the probability that two balls are of the same colour?

Two cards are drawn one after the other from a pack of 52 cards with replacement. What is the probability that both the cards drawn are face cards?

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 7 Probability Miscellaneous Exercise 7 [Pages 109 - 110]

From a group of 2 men (M_{1}, M_{2}) and three women (W_{1}, W_{2}, W_{3}), two persons are selected. Describe the sample space of the experiment. If E is the event in which one man and one woman are selected, then which are the cases favourable to E?

Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys and 1 girl and 3 boys. One child is selected at random from each group. What is the chance that three selected consists of 1 girl and 2 boys?

A room has three sockets for lamps. From a collection of 10 light bulbs of which 6 are defective, a person selects 3 bulbs at random and puts them in socket. What is the probability that the room is lit?

There are 2 red and 3 black balls in a bag. 3 balls are taken out at random from the bag. Find the probability of getting 2 red and 1 black ball or 1 red and 2 black balls.

A box contains 25 tickets numbered 1 to 25. Two tickets are drawn at random. What is the probability that the product on the numbers is even?

A, B, and C are mutually exclusive and exhaustive events associated with the random experiment. Find P(A), given that P(B) = `3/2` P(A) and P(C) = `1/2` P(B).

An urn contains four tickets marked with numbers 112, 121, 122, 222 and one ticket is drawn at random. Let A_{i} (i = 1, 2, 3) be the event that i^{th} digit of the number of the ticket drawn is 1. Discuss the independence of the events A_{1}, A_{2}, and A_{3}.

The odds against a certain event are 5: 2 and odds in favour of another independent event are 6: 5. Find the chance that at least one of the events will happen.

The odds against a husband who is 55 years old living till he is 75 is 8: 5 and it is 4: 3 against his wife who is now 48, living till she is 68. Find the probability that the couple will be alive 20 years hence.

The odds against a husband who is 55 years old living till he is 75 is 8: 5 and it is 4: 3 against his wife who is now 48, living till she is 68. Find the probability that at least one of them will be alive 20 years hence.

Two throws are made, the first with 3 dice and the second with 2 dice. The faces of each die are marked with the number 1 to 6. What is the probability that the total in first throw is not less than 15 and at the same time the total in the second throw is not less than 8?

Two-third of the students in a class are boys and rest are girls. It is known that the probability of girl getting first class is 0.25 and that of boy getting is 0.28. Find the probability that a student chosen at random will get first class.

A number of two digits is formed using the digits 1, 2, 3, ….., 9. What is the probability that the number so chosen is even and less than 60?

A bag contains 8 red balls and 5 white balls. Two successive draws of 3 balls are made without replacement. Find the probability that the first drawing will give 3 white balls and second drawing will give 3 red balls.

The odds against student X solving a business statistics problem are 8: 6 and odds in favour of student Y solving the same problem are 14: 16 What is the chance that the problem will be solved, if they try independently?

The odds against student X solving a business statistics problem are 8: 6 and odds in favour of student Y solving the same problem are 14: 16 What is the probability that neither solves the problem?

## Chapter 7: Probability

## Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board chapter 7 - Probability

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Concepts covered in Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board chapter 7 Probability are Introduction of Probability, Types of Events, Algebra of Events, Elementary Properties of Probability, Addition Theorem of Probability, Conditional Probability, Multiplication Theorem on Probability, Independent Events.

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